Existence Of Solutions For P(X)-laplace Problem On A Bounded Domain

In this paper, we study the existence of solutions for p ( x )-Laplace problem on a bounded domain. Our study is based on the basic theory of the variable exponent Lebesgue space, variable exponent Sobolev space, weighted variable exponent Lebesgue space and Sobolev space. In recent years, with the development of elastic mechanics, the study of variational problems with nonstandard growth conditions is an interesting topic.p ( x )-growth conditions can be regarded as a very important case of nonstandard growth conditions. Compared with p? Laplace problems, p ( x )-Laplace problems have more complex nonlinear nature. The critical point theory and mountain lemma play an important role in solving the p ( x )-Laplace problems of partial differential equations with nonstandard growth conditions. Using these tools, you can solve many problems about the existence of partial differential equations.In this paper, applying the basic theory of the variable exponent Lebesgue space, variable exponent Sobolev space, weighted variable exponent Lebesgue space and Sobolev space, we study the following p ( x )-Laplace problem: where 0?? is bounded domain in R^N. p:Ω→R^N is Lipschitz continuous and satisfies: 0＜b₀≤b(x)∈L^∞(Ω),0≤α∈C(?). Applying variational method, critical point theory, mountain lemma and the property of compaction, we obtain the existence of solutions in (?)for the p ( x )- Laplace problems in the superlinear and sublinear cases.